Finite Group Databasebeta

← #22599277

Subgroups — #22599277

Order 9088.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986554C211yesmaximal, characteristic, complemented
2#127×D71C211yesmaximal, characteristic
3#17986373C211yesmaximal, characteristic, complemented
4#17986346C211yesmaximal, characteristic, complemented
5#17986345C211yesmaximal, characteristic, complemented
6#17986197C214yesmaximal, complemented
7#17986148C214yesmaximal, complemented
8#17986050C211yesmaximal, characteristic, complemented
9#492×C71C211yesmaximal, characteristic, complemented
10C2×D4×D71—21no—
11C2×D4×D71—21no—
12C2×D4×D71C2212yes—
13C2×#11765852C2214yescomplemented
14#14556128C2211yescharacteristic, complemented
15#14556125C2211yescharacteristic
16#14556124—21no—
17#14556124—21no—
18#14556124C2212yes—
19#14556119—21no—
20#14556108—21no—
21#14556104C2212yes—
22Q8×Dic71C2212yescomplemented
23#14556093—21no—
24#14556086—21no—
25C42×D71C2211yescharacteristic
26Dic71⋊C8C2212yescomplemented
27#14556050C2214yescomplemented
28#14556049C2214yescomplemented
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic, complemented
31C2×SD16×C71C2214yescomplemented
32C4⋊Q8×C71C2211yescharacteristic, complemented
33#127×C71C2211yescharacteristic
34C4×C8×C71C2211yescharacteristic, complemented
35C23×D71—22no—
36C2×Dic142—21no—
37C2×D284—21no—
38C2×D284—21no—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43D4×D71—44no—
44D4×D71—44no—
45D4×D71—44no—
46C2×C4×D71D411yescharacteristic
47C2×C4×D71D411yescharacteristic
48C2×C4×D71C2311yescharacteristic
49#11765852—44no—
50C2×C71⋊C8C2312yes—
51C4⋊Dic71—21no—
52C4⋊Dic71—22no—
53C4⋊Dic71C2312yes—
54Dic71⋊C4—22no—
55(C2×D71)⋊C4—22no—
56C4×Dic71—22no—
57C4×Dic71D411yescharacteristic
58C4×Dic71D411yescharacteristic
59C4×Dic71C2311yescharacteristic
60C2×Q8×C71C2312yes—
61C2×D4×C71—21no—
62C2×D4×C71—21no—
63C2×D4×C71C2312yes—
64SD16×C71—44no—
65C2×C8×C71C2312yes—
66C4⋊C4×C71—21no—
67C4⋊C4×C71—21no—
68C42×C71C2311yescharacteristic
69C23×C71—22no—
70C22×D71—22no—
71C22×D71—44no—
72C22×D71—44no—
73C22×D71—44no—
74C22×D71C2×D411yescharacteristic
75Q8×C71—24no—
76D4×C71—24no—
77D4×C71—42no—
78D4×C71—42no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C22⋊D71—44no—
84C22⋊D71—44no—
85C22⋊D71—44no—
86C2×Dic71—22no—
87C2×Dic71C2×D411yescharacteristic
88C2×Dic71C2×D411yescharacteristic
89C2×Dic71C2×D411yescharacteristic
90D284—24no—
91D284—42no—
92D284—42no—
93C4×D71—22no—
94C4×D71—22no—
95C4×D71SD1614yescomplemented
96Dic142—42no—
97C8×C71—22no—
98C71⋊C8—22no—
99C22×C71—42no—
100C22×C71—42no—
101C22×C71—42no—
102C22×C71C22×D411yescharacteristic
103C2×D71—21no—
104C2×D71—21no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—88no—
109C2×D71C2×SD1612yes—
110C4×C71—42no—
111C4×C71C2×SD1612yes—
112C4×C71AGL(1,ℤ/8)12yes—
113C4×C71H(5,𝔽2)12yes—
114Dic71—21no—
115Dic71—21no—
116Dic71—42no—
117Dic71C2×SD1612yes—
118C2×C71—82no—
119C2×C71#53511yescharacteristic
120C2×C71C22×SD1611yescharacteristic
121C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
122D71—22no—
123D71—82no—
124#3208—711nomaximal, Sylow
125C71#320811yescharacteristic, complemented, Sylow
126#533—711no—
127C2×#127—711no—
128#492—711no—
129#492D7111yescharacteristic, complemented
130#486—711no—
131#486—711no—
132#460—714no—
133#450—714no—
134#431—711no—
135C22×D4—712no—
136C22×D4—1421no—
137C22×D4—1421no—
138C2×SD16—714no—
139C2×SD16C2×D7114yescomplemented
140C4⋊Q8—711no—
141C4⋊Q8C2×D7111yescharacteristic, complemented
142#127—712no—
143#127—711no—
144#127—1421no—
145#127—1421no—
146#127C2×D7111yescharacteristic
147#125—1421no—
148C22⋊Q8—712no—
149C22⋊Q8—1421no—
150C4×Q8—712no—
151C4×Q8—1421no—
152#117—1421no—
153C2×C42—711no—
154C4⋊C8—712no—
155#102—714no—
156#102—714no—
157C22⋊C8—712no—
158C4×C8—711no—
159C4×C8C2×D7111yescharacteristic, complemented
160C24—1422no—
161C2×Q8—1421no—
162C2×Q8C22×D7112yes—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—2844no—
166C2×D4—2844no—
167C2×D4—2844no—
168C2×D4—712no—
169C2×D4—1422no—
170C2×D4—1422no—
171C2×D4—1422no—
172C2×D4—1421no—
173C2×D4—1421no—
174C2×D4C22×D7112yes—
175C22×C4—711no—
176C22×C4—711no—
177C22×C4—711no—
178SD16—44no—
179SD16—2844no—
180C2×C8—712no—
181C2×C8C22×D7112yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—712no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1421no—
188C22⋊C4—1422no—
189C42—711no—
190C42—711no—
191C42—711no—
192C42—1422no—
193C42C22×D7111yescharacteristic
194C23—22no—
195C23—2844no—
196C23—2844no—
197C23—2844no—
198C23—711no—
199C23—1422no—
200Q8—24no—
201Q8—2842no—
202D4—24no—
203D4—42no—
204D4—42no—
205D4—2844no—
206D4—2844no—
207D4—2844no—
208D4—2842no—
209D4—2842no—
210D4—1424no—
211C2×C4—22no—
212C2×C4—711no—
213C2×C4—714no—
214C2×C4—711no—
215C2×C4—711no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4D4×D7111yescharacteristic
220C2×C4D4×D7111yescharacteristic
221C2×C4C23×D7111yescharacteristic
222C8—22no—
223C8—1422no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—2842no—
228C22—2842no—
229C22—2842no—
230C22—5688no—
231C22—712no—
232C22—1421no—
233C22—1421no—
234C22C2×D4×D7111yescharacteristic
235C4—42no—
236C4—2842no—
237C4—712no—
238C4—1421no—
239C4—1421no—
240C4SD16×D7112yes—
241C4#1455615512yes—
242C4#1455617312yes—
243C2—82no—
244C2—5682no—
245C2—1422no—
246C2#1798656211yescharacteristic
247C2C2×SD16×D7111yescharacteristic
248C2C2×#1455615511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#492×C71C2
socleC22×C71C22×D4
hypercenter#492D71
hypocenterC71#3208
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986373C2semidirect
4. #17986346C2semidirect
5. #17986345C2semidirect
6. #17986197C2semidirect
7. #17986148C2semidirect
8. #17986050C2semidirect
9. #492×C71C2semidirect
13. C2×#11765852C22semidirect
14. #14556128C22semidirect
22. Q8×Dic71C22semidirect
26. Dic71⋊C8C22semidirect
27. #14556050C22semidirect
28. #14556049C22semidirect
30. C4×C71⋊C8C22semidirect
31. C2×SD16×C71C22semidirect
32. C4⋊Q8×C71C22semidirect
34. C4×C8×C71C22semidirect
95. C4×D71SD16semidirect
125. C71#3208semidirect
129. #492D71semidirect
139. C2×SD16C2×D71semidirect
141. C4⋊Q8C2×D71semidirect
159. C4×C8C2×D71semidirect